Question: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 2x + 46$, and $ m \angle MON = 3x - 6$, find $m\angle MON$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 46} + {3x - 6} = {90}$ Combine like terms: $ 5x + 40 = 90$ Subtract $40$ from both sides: $ 5x = 50$ Divide both sides by $5$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 3({10}) - 6$ Simplify: $ {m\angle MON = 30 - 6}$ So ${m\angle MON = 24}$.